A Singularly Valuable Decomposition: The SVD of a Matrix
Dan Kalman
The American University
Washington, DC 20016
February 13, 2002
Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or
SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and
theoretical insights about linear transformations. The close connection between the SVD and the well
known theory of diagonalization for symmetric matrices makes the topic immediately accessible to linear
algebra teachers, and indeed, a natural extension of what these teachers already know. At the same
time, the SVD has fundamental importance in several different applications of linear algebra. Strang
was aware of these facts when he introduced the SVD in his now classical text [22, page 142], observing
. . . it is not nearly as famous as it should be.
Golub and Van Loan ascribe a central significance to the SVD in their definitive explication of numerical
matrix methods [8, page xiv] stating
. . . perhaps the most recurring theme in the book is the practical and theoretical value of [the
SVD].
Additional evidence of the significance of the SVD is its central role in a number of papers in recent
years in Mathematics Magazine and The American Mathematical Monthly (for example [2, 3, 17, 23]).
Although it is probably not feasible to include the SVD in the first linear algebra course, it definitely
deserves a place in more advanced undergraduate courses, particularly those with a numerical or applied
emphasis. My primary goals in this article are to bring the topic to the attention of a broad audience,
and to reveal some of the facets that give it both practical and theoretical significance.
1
Theory
The SVD is intimately related to the familiar theory of diagonalizing a symmetric matrix. Recall that
if A is a symmetric real n × n matrix, there is an orthogonal matrix V and a diagonal D such that
A = V DV T . Here the columns of V are eigenvectors for A and form an orthonormal basis for Rn ; the
diagonal entries of D are the eigenvalues of A. To emphasize the connection with the SVD, we will refer
to V DV T as the eigenvalue decomposition, or EVD, for A.
For the SVD we begin with an arbitrary real m × n matrix A. As we shall see, there are orthogonal
matrices U and V and a diagonal matrix, this time denoted Σ, such that A = U ΣV T . In this case, U is
m × m and V is n × n, so that Σ is rectangular with the same dimensions as A. The diagonal entries
of Σ, that is the Σii = σi , can be arranged to be nonnegative and in order of decreasing magnitude.
The positive ones are called the singular values of A. The columns of U and V are called left and right
singular vectors, for A.
The analogy between the EVD for a symmetric matrix and SVD for an arbitrary matrix can be
extended a little by thinking of matrices as linear transformations. For a symmetric matrix A, the
transformation takes Rn to itself, and the columns of V define an especially nice basis. When vectors
are expressed relative to this basis, we see that the transformation simply dilates some components and
contracts others, according to the magnitudes of the eigenvalues (with a reflection through the origin
tossed in for negative eigenvalues). Moreover, the basis is orthonormal, which is the best kind of basis
to have.
Now let’s look at the SVD for an m × n matrix A. Here the transformation takes Rn to a different
space, Rm , so it is reasonable to ask for a natural basis for each of domain and range. The columns
of V and U provide these bases. When they are used to represent vectors in the domain and range
of the transformation, the nature of the transformation again becomes transparent: it simply dilates
some components and contracts others, according to the magnitudes of the singular values, and possibly
discards components or appends zeros as needed to account for a change in dimension. From this
perspective, the SVD tells us how to choose orthonormal bases so that the transformation is represented
by a matrix with the simplest possible form, that is, diagonal.
How do we choose the bases {v1 , v2 , · · · , vn } and {u1 , u2 , · · · , um } for the domain and range? There
is no difficulty in obtaining a diagonal representation. For that, we need only Avi = σi ui , which is easily
arranged. Select an orthonormal basis {v1 , v2 , · · · , vn } for Rn so that the first k elements span the row
space of A and the remaining n − k elements span the null space of A, where k is the rank of A. Then for
1 ≤ i ≤ k define ui to be a unit vector parallel to Avi , and extend this to a basis for Rm . Relative to
these bases, A will have a diagonal representation. But in general, although the v’s are orthogonal, there
is no reason to expect the u’s to be. The possibility of choosing the v–basis so that its orthogonality is
preserved under A is the key point. We show next that the EVD of the n × n symmetric matrix AT A
provides just such a basis, namely, the eigenvectors of AT A.
Let AT A = V DV T , with the diagonal entries λi of D arranged in nonincreasing order, and let the
2
columns of V (which are eigenvectors of AT A) be the orthonormal basis {v1 , v2 , · · · , vn }. Then
Avi · Avj = (Avi )T (Avj ) = viT AT Avj = viT (λj vj ) = λj vi · vj
so the image set {Av1 , Av2 , · · · , Avn } is orthogonal, and the nonzero vectors in this set form a basis
for the range of A. Thus, the eigenvectors of AT A and their images under A provide orthogonal bases
allowing A to be expressed in a diagonal form.
To complete the construction, we normalize the vectors Avi . The eigenvalues of AT A again appear
in this step. Taking i = j in the calculation above gives |Avi |2 = λi , which means λi ≥ 0. Since these
eigenvalues were assumed to be arranged in nonincreasing order, we conclude that λ1 ≥ λ2 ≥ · · · ≥ λk ≥
0 and, since the rank of A equals k, λi = 0 for i > k. The orthonormal basis for the range is therefore
defined by
Avi
1
= √ Avi ; 1 ≤ i ≤ k
ui =
|Avi |
λi
If k < m, we extend this to an orthonormal basis for Rm .
√
This completes the construction of the desired orthonormal bases for Rn and Rm . Setting σi = λi
we have Avi = σi ui for all i ≤ k. Assembling the vi as the columns of a matrix V and the ui to form
U, this shows that AV = U Σ, where Σ has the same dimensions as A, has the entries σi along the
main diagonal, and has all other entries equal to zero. Hence, A = U ΣV T , which is the singular value
decomposition of A.
In summary, an m × n real matrix A can be expressed as the product U ΣV T , where V and U
are orthogonal matrices and Σ is a diagonal matrix, as follows. The matrix V is obtained from the
diagonal factorization AT A = V DV T , in which the diagonal entries of D appear in non-increasing
order; the columns of U come from normalizing the nonvanishing images under A of the columns of V,
and extending (if necessary) to an orthonormal basis for Rm ; the nonzero entries of Σ are the respective
square roots of corresponding diagonal entries of D.
The preceding construction demonstrates that the SVD exists, and gives some idea of what it tells
about a matrix. There are a number of additional algebraic and geometric insights about the SVD that
are derived with equal ease. Before proceeding to these insights, two remarks should be made. First,
the SVD encapsulates the most appropriate bases for the domain and range of the linear transformation
defined by the matrix A. There is a beautiful relationship between these bases and the four fundamental
subspaces associated with A: the range and nullspace, and their orthogonal complements. It is the full
picture provided by the SVD and these subspaces that Strang has termed the fundamental theorem of
linear algebra. He also invented a diagram schematically illustrating the relationship of the bases and
the four subspaces (see Fig. (1)). Strang’s article [23] is recommended for a detailed discussion of this
topic.
The second remark concerns computation. There is often a gap between mathematical theory and
computational practice. In theory, we envision arithmetic operations being carried out on real numbers
in infinite precision. But when we carry out arithmetic on a digital computer we compute not with reals,
but with a finite set of rationals, and the results can only approximate with limited precision the real
3
ra
e(
ng
)⊥
A)
A
ll(
nu
u1 ... uk
A
v1 ... vk
uk+1 ... um
vk+1 ... vn
A)
ll(
nu
⊥
A)
(
e
ng
ra
Figure 1: Strang’s Diagram
computations. Procedures that seem elegant and direct in theory can sometimes be dramatically flawed
as prescriptions for computational algorithms. The SVD illustrates this point. Our construction appears
to offer a straightforward algorithm for the SVD: form AT A, compute its eigenvalues and eigenvectors,
and then find the SVD as described above. Here practice and theory go their separate ways. As we
shall see later, the computation of AT A can be subject to a serious loss of precision. It turns out that
direct methods exist for finding the SVD of A without forming AT A, and indeed in many applications
the practical importance of the SVD is that it allows one to find the EVD of AT A without ever forming
this numerically treacherous product.
Let us now proceed to several additional aspects of the SVD.
The SVD and EVD. Our construction showed how to determine the SVD of A from the EVD
of the symmetric matrix AT A. Conversely, it is easy to recover the EVD of AT A from the SVD of
A. For suppose the singular value decomposition A = U ΣV T is given. Clearly, AT A = V ΣT ΣV T and
4
AAT = U ΣΣT U T . Now in either order the product of Σ and ΣT is a square diagonal matrix whose first k
diagonal entries are the σi2 , and with any remaining diagonal entries equal to 0. Thus, AT A = V ΣT ΣV T
is the EVD of AT A and and AAT = U ΣΣT U T is the EVD of AAT . Our argument also yields a
uniqueness result for the singular value decomposition. In any SVD of A, the right singular vectors
(columns of V ) must be the eigenvectors of AT A, the left singular vectors (columns of U ) must be the
eigenvectors of AAT , and the singular values must be the square roots of the nonzero eigenvalues common
to these two symmetric matrices. Thus, up to possible orthogonal transformations in multidimensional
eigenspaces of AT A and AAT , the matrices V and U in the SVD are uniquely determined. Finally, note
that if A itself is square and symmetric, each eigenvector for A with eigenvalue λ is an eigenvector for
A2 = AT A = AAT with eigenvalue λ2 . Hence the left and right singular vectors for A are simply the
eigenvectors for A, and the singular values for A are the absolute values of its eigenvalues. That is,
the EVD and SVD essentially coincide for symmetric A, and are actually identical if A has no negative
eigenvalues. In particular, for any A, the SVD and EVD of AT A are the same.
A geometric interpretation of the SVD. One way to understand how A deforms space is to
n
consider its action on the unit sphere in R
arbitrary element x of this unit sphere can be represented
Pn. An
by x = x1 v1 + x2 v2 + · · · + xn vn with 1 x2i = 1. The image is Ax = σ1 x1 u1 + · · · + σk xk uk . Letting
yi = xi σi , we see the image of the unit sphere consists of the vectors y1 u1 + y2 u2 + · · · + yk uk , where
y2
y2
y12
+ 22 + · · · + k2
2
σ1
σ2
σk
=
k
X
x2i
1
≤ 1
If A has full column rank, so that k = n, the inequality is actually a strict equality. Otherwise, some
of the xi are missing on the right, and the sum can be anything from 0 to 1. This shows that A maps
the unit sphere of Rn to a k–dimensional ellipsoid with semi-axes in the directions ui and with the
magnitudes σi . If k = n the image is just the surface of the ellipsoid, otherwise it is the solid ellipsoid.
In summary, we can visualize the effect A as follows: it first collapses n − k dimensions of the domain,
then distorts the remaining dimensions, stretching and squeezing the unit k–sphere into an ellipsoid,
and finally embeds the ellipsoid in Rm . This is illustrated for n = m = 3 and k = 2 in Fig. (2).
As an immediate consequence, we see that kAk, the operator norm of A, defined as the maximum
value of |Av| for v on the unit sphere, is simply σ1 , the largest singular value of A. Put another way, we
have the inequality |Ax| ≤ σ1 |x| for all x ∈ Rn , with equality only when x is a multiple of v1 .
Partitioned matrices and the outer product form of the SVD. When viewed in a purely
algebraic sense, any zero rows and columns of the matrix Σ are superfluous. They can be eliminated if
5
v3
v1
σ1u1
v2
σ2 u2
v2
v1
Figure 2: How A deforms Rn
the matrix product A = U ΣV T is expressed using partitioned matrices as follows

  vT
1
σ1
 ..


.
..

0   .T
¤
£

  vk
σk
A = u1 · · · uk uk+1 · · · um 
 T

  vk+1



0
0   ...










vnT
Although these partitions assume that k is strictly less than m and n, it should be clear how to modify
the arguments if k is equal to m or n. When the partitioned matrices are multiplied, the result is


 T 
 T 
vk+1
σ1
v1
¤
¤
£
£





.
. 
.
..
A = u 1 · · · uk 
  ..  + uk+1 · · · um 
  .. 
σk
vkT
vnT
0
From this last equation it is clear that only the first k u’s and v’s make any contribution to A.
Indeed, we may as well shorten the equation to

 T 
σ1
v1
¤
£
  .. 
.
.
A = u 1 · · · uk 
 . 
.
σk
vkT
Notice that in this form the matrices of u’s and v’s are now rectangular (m × k and k × n respectively),
and the diagonal matrix is square. This is an alternate version of the SVD that is taken as the definition
6
in some expositions: Any m × n matrix A of rank k can be expressed in the form A = U ΣV T where U
is an m × k matrix such that U T U = I, Σ is a k × k diagonal matrix with positive entries in decreasing
order on the diagonal, and V is an n × k matrix such that V T V = I.
The partitioned matrix formulation of the SVD is a little unusual in one respect. Usually in a matrix
product XY , we focus on the rows in X and on the columns in Y . Here, the factors are expressed in just
the opposite way. This is the ideal situation to apply what is called an outer product expansion for the
product of two matrices. In general, if X is an m × k matrix with columns xi and Y is a k × n matrix
with rows yiT , the matrix product XY can be expressed as
XY =
k
X
xi yiT
i=1
Each of the terms xi yiT is an outer product of vectors xi and yj . It is simply the standard matrix product
of a column matrix and a row matrix. The result can be visualized in terms of a multiplication table,
with the entries of xi listed along the left margin and those of yj across the top. The body of the table is
the outer product of xi and yj . This idea is illustrated in Fig. (3), showing
h ithe outer product of (a, b, c)
and (p, q, r). Observe that in the figure, each column is a multiple of
p
a ap
b bp
c cp
q
aq
bq
cq
a
b
c
and each row is a multiple
r
ar
br
cr
Figure 3: Outer Product as Multiplication Table
£
¤
of p q r , so that the outer product is clearly of rank 1. In just the same way, the outer product
xi yiT is a rank 1 matrix with columns which are multiples of xi and rows which are multiples of yi .
We shall return to outer product expansions in one of the applications of the SVD. Here, we simply
apply the notion to express the SVD of A in a different form.

Let
X=
£
u1
· · · uk
¤


σ1
..
 £
 = σ1 u1
.
σk
and

v1T


Y =  ... 
vkT

7
· · · σk u k
¤
Then A = XY can be expressed as an outer product expansion
A=
k
X
σi ui viT
i=1
This is yet another form of the SVD, and it provides an alternate way of expressing how A transforms
an arbitrary vector x. Clearly,
k
X
σi ui viT x
Ax =
i=1
Since
viT x
is a scalar, we can rearrange the order of the factors to
Ax =
k
X
viT xσi ui
i=1
Now in this sum Ax is expressed as a linear combination of the vectors ui . Each coefficient is a product of
two factors, viT x and σi . Of course, viT x = vi · x is just the ith component of x relative to the orthonormal
basis {v1 , · · · , vn }. Viewed in this light, the outer product expansion reconfirms what we already know:
under the action of A each v component of x becomes a u component after scaling by the appropriate σ.
Applications
Generally, the SVD finds application in problems involving large matrices, with dimensions that can reach
into the thousands. It is the existence of efficient and accurate computer algorithms for its computation
that makes the SVD so useful in these applications. There is beautiful mathematics involved in the
derivation of the algorithms, and the subject is worth looking into. However, for the present discussion
of applications, we will treat the computation of the SVD as if performed by a black box. By way of
analogy, consider any application of trigonometry. When we require the value of a sine or cosine, we
simply push the buttons on a calculator with never a thought to the internal workings. We are confident
that the calculator returns a close enough approximation to serve our purposes, and think no more
about it. So, too, in the present discussion, let the reader be confident that computers can quickly and
accurately approximate the SVD of arbitrary matrices. To extend the analogy, we will concentrate on
when and why to push the SVD button, and how to use the results.
The SVD is an important tool in several different applications. Here we will present a detailed
discussion of two samples that are apparently quite unrelated: linear least squares optimization and
data compression with reduced rank approximations. Before proceeding to these topics, a few other
applications will be briefly mentioned.
Applications in Brief. One of the most direct applications of the SVD is to the problem of computing
the EVD of a matrix product AT A. This type of problem is encountered frequently under the name of
8
principal components analysis1 , and in statistics in connection with the analysis of covariance matrices2 .
As the discussion of least squares optimization will make clear, the computation of AT A can lead to a
significant degradation of accuracy of the results. In contrast, the SVD can be computed by operating
directly on the original matrix A. This gives the desired eigenvectors of AT A (the right singular vectors
of A) and eigenvalues of AT A (the squares of the singular values of A) without ever explicitly computing
AT A.
As a second application, the SVD can be used as a numerically reliable estimate of the effective rank
of the matrix. Often linear dependencies in data are masked by measurement error. Thus, although
computationally speaking the columns of a data matrix appear to be linearly independent, with perfect
measurement the dependencies would have been detected. Or, put another way, it may be possible to
make the columns of the data matrix dependent by perturbing the entries by small amounts, on the
same order as the measurement errors already present in the data. The way to find these dependencies
is to focus on the singular values that are of a larger magnitude than the measurement error. If there
are r such singular values, the effective rank of the matrix is found to be r. This topic is closely related
to the idea of selecting the closest rank r approximation to a matrix, which is further considered in the
discussion of data compression.
A third application of the SVD is to the computation of what is called the generalized inverse of a
matrix. This is very closely related to the linear least squares problem, and will be mentioned again in
the discussion of that topic. However, since the subject of generalized inverses is of interest in its own
right, it deserves brief mention here among the applications of the SVD.
This concludes the list of briefly mentioned applications. Next we proceed to take a detailed look
at least squares problems.
Linear Least Squares. The general context of a linear least squares problem is this: we have a set of
vectors which we wish to combine linearly to provide the best possible approximation to a given vector.
If the set of vectors is {a1 , a2 , · · · , an } and the given vector is b, we seek coefficients x1 , x2 , · · · , xn which
produce a minimal error
¯
¯
n
¯
¯
X
¯
¯
xi ai ¯
¯b −
¯
¯
i=1
The problem can arise naturally in any vector space, with elements which are sequences, functions,
solutions to differential equations, and so on. As long as we are only interested in linear combinations
of a finite set {a1 , a2 , · · · , an }, it is possible to transform the problem into one involving finite columns
of numbers. In that case, define a matrix A with columns given by the ai , and a vector x whose entries
are the unknown coefficients xi . Our problem is then to choose x minimizing |b − Ax|. As in earlier
discussions, we will denote the dimensions of A by m and n, meaning in this context that the ai are
vectors of length m.
1 An application to digital image processing is described in an introductory section of [15, Chapter 8]. Succeeding
sections discuss the SVD and its other applications. A much more detailed presentation concerning image processing
appears in [21, Chapter 6].
2 See, for example, [9].
9
There is a geometric interpretation to the general least squares problem. We are seeking an element
of the subspace S spanned by the ai which is closest to b. The solution is the projection of b on S, and
is characterized by the condition that the error vector (that is, the vector difference between b and its
projection) should be orthogonal to S. Orthogonality to S is equivalent to orthogonality to each of the
ai . Thus, the optimal solution vector x must satisfy ai · (Ax − b) = 0 for all i. Equivalently, in matrix
form, AT (Ax − b) = 0.
Rewrite the equation as AT Ax = AT b, a set of equations for the xi generally referred to as the
normal equations for the linear least squares problem. Observe that the independence of the columns of
A implies the invertibility of AT A. Therefore, we have x = (AT A)−1 AT b.
That is a beautiful analysis. It is neat, it is elegant, it is clear. Unfortunately it is also poorly
behaved when it is implemented in approximate computer arithmetic. Indeed, this is the classic example
of the gap between theory and practice alluded to earlier. Numerically, the formation of AT A can
dramatically degrade the accuracy of a computation; it is a step to be avoided. A detailed discussion of
the reasons for this poor performance is provided in [8]. Here, we will be content to give some insight
into the problem, using both a heuristic analysis and a numerical example. First, though, let us see how
the SVD solves the least squares problem.
We are to choose the vector x so as to minimize |Ax − b|. Let the SVD of A be U ΣV T (where U
and V are square orthogonal matrices, and Σ is rectangular with the same dimensions as A). Then we
have
Ax − b
= U ΣV T x − b
= U (ΣV T x) − U (U T b)
= U (Σy − c)
where y = V T x and c = U T b. Now U is an orthogonal matrix, and so preserves lengths. That is,
|U (Σy − c)| = |Σy − c|, and hence |Ax − b| = |Σy − c|. This suggests a method for solving the least
squares problem. First, determine the SVD of A and calculate c as the product of U T and b. Next, solve
the least squares problem for Σ and c. That is, find a vector y so that |Σy − c| is minimal. (We shall see
in a moment that the diagonal nature of Σ makes this step trivial.) Now y = V T x so we can determine
x as V y. That gives the solution vector x as well as the magnitude of the error, |Σy − c|.
In effect, the SVD has allowed us to make a change of variables so that the least squares problem is
reduced to a diagonal form. In this special form, the solution is easily obtained. We seek y to minimize
the norm of the vector Σy − c. Let the nonzero diagonal entries of Σ be σi for 1 ≤ i ≤ k, and let the
10
components of y, which is the unknown, be yi for 1 ≤ i ≤ n. Then



σ1 y1 − c1
σ 1 y1

 .. 
..

 . 
.



 σk yk − ck
 σk yk 





Σy = 
 0  so Σy − c =  −ck+1
 −ck+2
 0 




 . 
..

 .. 
.












−cm
0
By inspection, when yi = ci /σi for 1 ≤ i ≤ k, Σy − c assumes its minimal length, which is given by
"
m
X
#1/2
c2i
(1)
i=k+1
Note that when k = m, the sum is vacuous. In this case, the columns of A span Rm so the least squares
problem can be solved with zero error. Also observe that when k is less than n, there is no constraint
on the values of yk+1 through yn . These components can be assigned arbitrary values with no effect on
the length of Σy − c.
As the preceding analysis shows, the SVD allows us to transform a general least squares problem
into one that can be solved by inspection, with no restriction on the rank of the data matrix A. Indeed,
we can combine the transformation steps and the solution of the simplified problem as follows. First,
we know that c = U T b. The calculation of y from c amounts to multiplying by the matrix Σ+ defined
by transposing Σ and inverting the nonzero diagonal entries. Then y = Σ+ c will have its first k entries
equal to ci /σi , as required. Any remaining entries (which were previously unconstrained) all vanish.
Finally, we compute x = V y. This gives the compact solution
x = V Σ+ U T b
(2)
At this point, the subject of generalized inverses surfaces. Given b ∈ Rm , there is a unique element
x ∈ Rn of minimal norm for which Ax is nearest b. The mapping from b to that x defines what is called
the Moore-Penrose generalized inverse, A+ , of A. It is clear that Σ+ , as defined in the solution of the
lease squared problem for Σ and c, is the Moore-Penrose inverse of Σ. Moreover, Eq. (2) shows that when
A = U ΣV T is the SVD of A, the generalized inverse is given by A+ = V Σ+ U T . A nice treatment of the
generalized inverse that is both compact and self-contained is given in [5], although there is no mention
of the SVD in that work. Generalized inverses and their connection with the SVD are also discussed in
[8, 14, 22, 23].
Now we are in a position to compare the solutions obtained using the SVD and the normal equations.
The solutions are theoretically equivalent, so if they are carried out in exact arithmetic, they must yield
the same results. But when executed in limited precision on a computer, they can differ significantly.
11
At the heart of this difference lie the effects of roundoff error. Accordingly, before proceeding, we pause
to consider one important aspect of roundoff error.
An example will make this aspect clear. Let x = 1.23456789012345 · 108 and y = 1.23456789012345 ·
10 . Computing with 15 correct digits, the sum of x and y is given by
−1
12345678
+
12345679
. 9012345
. 123456789012345
. 0246912
which corresponds to introducing an error in about the 8th decimal place of y. Observe that while each
number is represented in the computer with a precision of 15 decimal places, the least significant digits
of one term are essentially meaningless. If the two terms differ in magnitude by a sufficient amount (16
orders of magnitude in the example), the contribution of the lesser term is completely eliminated.
This problem shows up in the inversion of linear systems when the system matrix has singular values
that vary widely in magnitude. As an example, consider the inversion of AT A that occurs as part of
the normal equations. The normal equations formulation requires that AT A be nonsingular. If nearly
singular, AT A will have a singular value very near 0, and this is when the normal equations become
unreliable. Let us write the singular values of AT A in decreasing order as λ1 , λ2 , · · · , λn . These are
actually the eigenvalues of AT A (as observed earlier) and they are the squares of the singular values of
A. The eigenvectors of AT A are v1 , v2 , · · · , vn , the right singular vectors of A. With λn very near 0, it
is likely that the λi will cover a wide range of magnitudes.
To dramatize this situation, we will assume that λn is several orders of magnitude smaller than
λ1 , and that the other λi are roughtly comparable to λ1 in magnitude. Geometrically, AT A maps the
unit sphere of Rn onto an ellipsoid with axes in the direction of the vi . The length of the axis in the
direction of vn is so much smaller than all the other axes that the ellipsoid appears to completely flatten
out and lie in an n − 1 dimensional hyperplane (see Fig. (4)). Therefore, in the image under AT A of a
ATA
Figure 4: Unit Sphere Deformed by AT A
random vector of unit length, the expected contribution of the vn component is essentially negligable.
In this case, the effect of finite precision is to introduce a significant error in the contribution of vn .
12
However, in solving the normal equations we are interested in the inverse mapping, (AT A)−1 , which has
the same eigenvectors as AT A, but whose eigenvalues are the reciprocals 1/λi . Now it is λn , alone, that
is significant, and the ellipsoid is essentially one dimensional (see Fig. (5)). In every direction except vn ,
the image of a random unit vector will have no noticable contribution. Arguing as before, significant
(ATA)-1
Figure 5: Unit Sphere Deformed by (AT A)−1
errors are introduced in the contributions of all of the vi except vn .
Let us make this analysis more concrete. Let a particular vector x be selected, with x = x1 v1 +
x2 v2 + · · · + xn vn . Thinking of x as a column vector, consider a single entry, c, and write ci for the
corresponding entry of xi vi . Then c = c1 + c2 + · · · + cn . In the image (AT A)−1 x the corresponding entry
will be c1 /λ1 + c2 /λ2 + · · · + cn /λn . Now if the ci are all of roughly comparable magnitude, then the
final term will dwarf all the rest. The situation illustrated above will occur. The cn /λn term will play
the role of 12345678.9012345; every other term will be in the position of .123456789012345. In effect,
several decimal digits of each term, save the last, are neglected. Of course, the number of neglected
digits depends on the difference in magnitude between λ1 and λn . If λ1 /λn is on the order of 10τ , then
λ1 and λn differ by τ orders of magnitude, and the inferior terms of the sum lose τ digits of accuracy.
If τ is large enough, the contributions can be completely lost from every term except the last.
Does this loss of accuracy really matter? Isn’t the value of y = (AT A)−1 x still correct to the accuracy
of the computation? If y is really all we care about, yes. But y is supposed to be an approximate
solution of AT Ay = x, and we should judge the adequacy of this approximation by looking at the
difference between AT Ay and x. In exact arithmetic, the first n − 1 ci /λi , which were nearly negligable
in the computation of y, regain their significance and make an essential contribution to the restoration
of x. But the digits that were lost in the limited precision arithmetic cannot be recovered, and so every
component, except the last, is either corrupted or lost entirely. Thus, while y = (AT A)−1 x might be
correct to the limits of computational precision, the computed value of AT Ay can be incorrect in every
decimal place.
In the context of the normal equations, the computation of x = (AT A)−1 AT b is supposed to result
13
in a product Ax close to b. This calculation is subject to the same errors as those described above.
Small errors in the computation of (AT A)−1 can be inflated by the final multiplication by A to severely
degrade the accuracy of the final result.
Based on the foregoing, it is the range of the magnitudes of the eigenvalues of AT A that determines
the effects of limited precision in the computation of the least-squares vector x, and these effects will
be most severe when the ratio λ1 /λn is large. As many readers will have recognized, this ratio is the
condition number of the matrix AT A. More generally, for any matrix, the condition number is the ratio
of greatest to least singular values. For a square matrix A, the condition number can be interpreted
as the reciprocal of the distance to the nearest singular matrix ([8, p. 26]). A large condition number
for a matrix is a sign that numerical instability may plague many types of calculations, particularly the
solution of linear systems.
Now all of this discussion applies equally to the SVD solution of the least squares problem. There
too we have to invert a matrix, multiplying U T b by Σ+ . Indeed, the singular values are explicitly
inverted, and we can see that a component in the direction of the smallest positive singular value gets
inflated, in the nonsingular case by the factor 1/σn . Arguing as before, the effects of truncation are felt
when the condition number σ1 /σn is large. So the benefit of the SVD solution is not that the effects of
near dependence of the columns of A are avoided. But let us compare the two condition numbers. We
know that the eigenvalues of AT A are the squares of the singular values of A. Thus λ1 /λn = (σ1 /σn )2 ,
the condition number of AT A is the square of the condition number of A. In view of our heuristic error
analysis, this relation between the condition numbers explains a rule of thumb for numerical computation:
the computation using AT A needs to be performed using roughly twice as many digits, to assure a final
result with accuracy comparable to that provided by the SVD of A.
The heuristic analysis is plausible, but by itself cannot be completely convincing. The following
numerical example provides some evidence in support of the heuristic argument. Of course, there is no
substitute for a careful mathematical analysis, and the interested reader is encouraged to consult [8] for
a discussion of condition number.
A Numerical Example We will simulate a least squares analysis where experimental data have been
gathered for four variables and we are attempting to estimate one, the dependent variable, by a linear
combination of the other three independent variables. For simplicity we suppose each variable is specified
by just four data values, so the columns a1 , a2 , a3 of A, as well as the dependent column b, lie in R4 .
We would like the columns of A to be nearly dependent, so that we can observe the situation described
in the heuristic analysis. This will be accomplished by making one column, say a3 , differ from a linear
combination of the other two by a very small random vector. Similarly, we want the dependent vector
b to be near, but not in, the range of A, so we will define it, too, as a linear combination of a1 and a2
plus a small random vector. We then compute the least squares coefficients x = (x1 , x2 , x3 ) by the two
formulas x = V Σ+ U T b and x = (AT A)−1 AT b, and compare the errors by calculating the magnitude of
the residual, |b − Ax|, in each case.
In choosing the random vectors in the definitions of a3 and b, we have different goals in mind. The
14
random component of b will determine how far b is from the range of A, and hence how large a residual
we should observe from a least squares solution. All our computations will be performed with 16 decimal
place accuracy. With that in mind, we will take the random component of b to be fairly small, on the
order of 10−4 . In contrast, the goal in choosing the random component of a3 is to make the condition
number of A on the order of 108 . Then the condition number of AT A will be on the order of 1016 , large
enough to corrupt all 16 decimal digits of the solution produced by the normal equations. As will be
shown in our later discussion of reduced rank approximations, the smallest singular value of A will be of
about the same magnitude as the norm of the random vector used to define a3 . If the norms of a1 and
a2 are on the order of 10, then the largest singular value will be, as well. Thus, choosing the random
component of b with norm on the order of 10−7 should produce a condition number in the desired range.
It is worth noting at this point that the near dependence between the columns of A should not
degrade the quality of the least squares solution. Using just the first two columns of A, it will be
possible to approximate b to within an error of about 10−4 . The third column of A can only improve the
approximation. So, the true least squares solution should be expected to produce a residual no greater
than 10−4 . The large error obtained using the normal equations is therefore attributable entirely to the
numerical instability of the algorithm, and not to any intrinsic limitations of the least squares problem.
All of the computations in this example were performed using the computer software package MATLAB [18]. Its simple procedures for defining and operating on matrices makes this kind of exploration
almost effortless. The reader is encouraged to explore this topic further, using MATLAB to recreate and
modify the example presented here. To make that process simpler, the presentation of the example will
include the exact MATLAB commands that were used at each step. When actually using the program,
the results of each computation are shown on the computer screen, but here, to save space, we show
only the final results.
First, define two data vectors:
c1 = [1 2 4 8]’
c2 = [3 6 9 12]’
(The primes indicate the transpose operator, so that c1 and c2 are defined as column vectors.) The
third data vector is a combination of these, plus a very small random vector:
c3 = c1 - 4*c2 + .0000001*(rand(4,1) - .5*[1 1 1 1]’)
and the matrix A is defined to have these three vectors as its columns:
A=[c1 c2 c3]
In the definition of c3 the command rand(4,1) returns a 4-entry column vector with entries randomly chosen between 0 and 1. Subtracting 0.5 from each entry shifts the entries to be between -1/2
15
and 1/2. Thus, the length of the random component of c3 will be at most 10−7 , in accord with the
discussion above.
Next, we define the b vector in a similar way, by adding a small random vector to a specified linear
combination of columns of A.
b = 2*c1 - 7*c2 + .0001*(rand(4,1) - .5*[1 1 1 1]’)
This time the random vector will have length no greater than 10−4 , again reflecting the previous discussion.
The SVD of A is quickly determined by MATLAB:
[U,S,V] = svd(A)
The three matrices U, S (which represents Σ), and V are displayed on the screen and kept in the computer
memory. The singular values when we ran the program turned out to be 59.810, 2.5976 and 1.0578×10−8 .
The condition number of A is therefore about 6 · 109 , and has the desired magnitude.
To compute the matrix S+ we need to transpose the diagonal matrix S and invert the nonzero
diagonal entries. This matrix will be denoted by G. It is defined by the following MATLAB commands:
G = S’
G(1,1) = 1/S(1,1)
G(2,2) = 1/S(2,2)
G(3,3) = 1/S(3,3)
Now let’s see how well the SVD solves the least squares problem. We multiply x = V Σ+ U T b by the
matrix A and see how far the result is from b. Upon receiving the commands
r1 = b - A*V*G*U’*b
e1 = sqrt(r1’*r1)
MATLAB responds
e1 = 4.2681e-05
As desired, the computed magnitude of the residual |b − Ax| is a bit smaller than 10−4 . So the SVD
provided a satisfactory solution of our least squares problem.
The solution provided by the normal equations is x = (AT A)−1 AT b. We enter
16
r2 = b - A*inv(A’*A)*A’*b
e2 = sqrt(r2’*r2)
and MATLAB responds
e2 = 51.9255
which is of the same order of magnitude as |b|, the distance from b to the origin! As we anticipated from
our heuristic analysis, the solution to the least squares problem computed using the normal equations
does a poor job of approximating b as a linear combination of the columns of A. What is more, the
computed residual for this method is of no value as an estimate of the distance of b from the range of A.
This completes our in depth look at least squares problems. The next section provides a detailed
look at another area of application: reduced rank approximation.
Data compression using reduced rank approximations. The expression of the SVD in terms of
the outer product expansion was introduced earlier. This representation emerges in a natural way in
another application of the SVD – data compression. In this application we begin with an m × n matrix
A of numerical data, and our goal is to describe a close approximation to A using many fewer numbers
than the mn original entries. The matrix is not considered as a linear transformation, or indeed as an
algebraic object at all. It is simply a table of mn numbers and we would like to find an approximation
that captures the most significant features of the data.
Because the rank of a matrix specifies the number of linearly independent columns (or rows), it is a
measure of redundancy. A matrix of low rank has a large amount of redundancy, and so can be expressed
much more efficiently than simply by listing all the entries. As a graphic example, suppose a scanner is
used to digitize a photograph, replacing the image by an m × n matrix of pixels, each assigned a gray
level on a scale of 0 to 1. Any large-scale features in the image will be reflected by redundancy in the
columns or rows of pixels, and thus we may hope to recapture these features in an approximation by a
matrix of lower rank than min(m, n).
The extreme case is a matrix of rank one. If B is such a matrix, then the columns are all multiples
of one another – the column space is one dimensional. If u is the single element of a basis, then each
column is a multiple of u. We represent the coefficients as vi , meaning that the ith column of B is given
by vi u, so that B = [v1 u v2 u · · · vn u] = uv T . Thus, any rank one matrix can be expressed as an outer
product, that is, as the product of a column and row. The mn entries of the matrix are determined
by the m entries of the column and the n entries of the row. For this reason, we can achieve a large
compression of the data matrix A if it can be approximated by a rank one matrix. Instead of the mn
entries of A, we need only m + n numbers to represent this rank one approximation to A. It is natural
to seek the best rank one approximation.
We will call B the best rank one approximation to A if the error matrix B − A has minimal norm,
where now we define the (Frobenius) norm |X| of a matrix X to be simply the square root of the sum of
17
the squares of its entries. This norm is just the Euclidean norm of the matrix considered as P
a vector in
Rmn . Thinking of matrices this way, the inner product of two matrices is defined by X · Y = ij xij yij ,
and as usual, |X|2 = X · X. Evidently this inner product of matrices can be thought of in three ways:
as the sum of the inner products of corresponding rows, the sum of the inner products of corresponding
columns, or as the sum of the mn products of corresponding entries.
There is a simple expression for the norm of a matrix product XY that is easily derived using the
outer product expansion. First, note that for rank one matrices xy T and uv T ,
X
X
xyi · uvi =
(x · u)yi vi = (x · u)(y · v)
(3)
xy T · uv T = [xy1 · · · xyn ] · [uv1 · · · uvn ] =
i
i
where we have computed the matrix inner product as the sum of vector inner products of corresponding
columns. In particular, xy T and uv T will be orthogonal with respect to the matrix inner inner product
provided that either
P x and u or y and v are orthogonal as vectors. Using Eq. (3) and the outer product
expansion XY = i xi yiT (in which the xi are the columns of X and the yiT are the rows of Y ) gives

! 
Ã
X
X
X
xi yiT · 
xj yjT  =
(xi · xj )(yi · yj )
XY · XY =
i
or
|XY |2 =
j
X
ij
|xi |2 |yi |2 +
i
X
(xi · xj )(yi · yj )
i6=j
As a special case, if the xi are orthogonal, then
|XY |2 =
X
|xi |2 |yi |2 .
i
Furthermore, if the xi are both orthogonal and of unit length,
X
|yi |2 = |Y |2 .
|XY |2 =
i
Similar conclusions apply if the yi are orthogonal, or orthonormal. This shows that the norm of a matrix
is unaffected by multiplying on either the left or right by an orthogonal
P matrix. Applying these results
to the outer product expansion for the SVD, observe that in A = P σi ui viT , the terms are orthogonal
|σi ui viT |2 so that the SVD can be
with respect to the matrix inner product. This shows that |A|2 =
3
viewed as an orthogonal decomposition of A into rank one matrices . More generally, if we partition
the sum, taking Sr as the sum of the first r terms and Er as the sum of the remaining terms, then
|A|2 = |Sr |2 + |Er |2 .
Now we are in a position to show a connection between rank one approximation and the SVD.
Specifically, we will show that the best rank one approximation to A must be of the form σ1 u1 v1T . Clearly,
P 2
3
2
This also shows that |A| =
σi . Interestingly, both the operator norm kAk = σ1 and the Frobenius norm |A| are
simply expressed in terms of singular values.
18
σ1 u1 v1T is one possible rank one approximation, and it gives rise to an error of σ22 + · · · + σk2 . The idea is
to show that this is the smallest the error can be. For any rank one A1 , using the fact that the Frobenius
norm is preserved under multiplication by orthogonal matrices, |A−A1 | = |U ΣV T −A1 | = |Σ−U T A1 V |,
where A = U ΣV T expresses the SVD of A. Write U T A1 V as αxy T with positive α and unit vectors
x ∈ Rm and y ∈ Rn . Next use the properties of the matrix inner product:
|Σ − αxy T |2 = |Σ|2 + α2 − 2αΣ · xy T
Focusing on the final term, the matrix inner product of the diagonal matrix Σ with the outer product
xy T is given by
Σ · xy T
=
k
X
σi xi yi
i=1
≤
k
X
σi |xi ||yi |
i=1
≤ σ1
k
X
|xi ||yi |
i=1
∗
= σ1 x · y ∗
where x∗ = (|x1 |, · · · , |xk |) and y ∗ = (|y1 |, · · · , |yk |). By the Cauchy–Schwarz inequality, x∗ · y ∗ ≤
|x∗ ||y ∗ | ≤ |x||y| = 1. Combined with the last equation above, this gives Σ · xy T ≤ σ1 .
Returning to the main thread of the argument, we now have |Σ − αxy T |2 ≥ |Σ|2 + α2 − 2ασ1 =
|Σ| + (α − σ1 )2 − σ12 . We immediately see that the right-hand side is minimized when α is taken to be
σ1 , so that |Σ − αxy T |2 ≥ |Σ|2 − σ12 . Moreover, if the σi are distinct, this minimum is actually obtained
only when α = σ1 and x and y are of the form e1 = (1, 0, · · · , 0) in Rm and Rn , respectively. Finally,
A1 = αU xy T V T = σ1 (U e1 )(V e1 )T = σ1 u1 v1T . That is what we wished to show. (If σ1 is a repeated
singular value, a slight modification of the argument is needed. It amounts to making an orthogonal
change of basis in the subspace of right singular vectors corresponding to σ1 .)
2
This result shows that the SVD can be used to find the best rank one approximation to a matrix. But
in many cases, the approximation will be too crude for practical use in data compression. The obvious
next step is to look for good rank r approximations for r > 1. In this regard, there is an attractive greedy
algorithm. To begin, choose a rank one A1 for which the error E1 = A − A1 has minimal norm. Next
choose a rank one matrix A2 for which the norm of E2 = E1 − A2 is minimal. Then A1 + A2 is a rank
two approximation to A with error E2 . Continue in this fashion, each time choosing a best possible rank
one approximation to the error remaining from the previous step. The procedure is a greedy algorithm
because at each step we attempt to capture as large a piece of A as possible. After r steps, the sum
A1 + · · · + Ar is a rank r approximation to A. The process can be repeated for k steps (where k is the
rank of A), at which point the error is reduced to zero. This results in the decomposition
X
Ai
A=
i
19
which is none other than the SVD of A. To be more precise, each Ai can be expressed as the product
of a positive scalar σi with an outer product ui viT of unit vectors. Then A is given by
X
σi ui viT
A=
i
which, true to the notation, is the SVD of A expressed in the form of the outer product expansion.
This statement follows from the earlier result linking the best rank one approximation of a matrix to
its largest singular
value and corresponding singular vectors. Assuming the singular
Pk values are distinct,
Pk
write A = i=1 σi ui viT . It is immediate that A1 must equal σ1 u1 v1T , and E1 = i=2 σi ui viT . But that
gives the SVD of E1 , from which we can instantly obtain the best rank one approximation A2 = σ2 u2 v2T .
Clearly, this argument can be repeated until the complete decomposition of A is obtained. (As before,
if the singular values of A are not distinct, the argument must be modified slightly to take into account
orthogonal changes of basis in the subspace of right singular vectors corresponding to a particular singular
value.)
The understanding we now have of the connection between successive best rank one estimates
and
P
the SVD can be summarized as follows. The outer product expansion form for the SVD, A = σi ui viT ,
expresses A as a sum of rank one matrices that are orthogonal with
Pr respect to the matrix inner product.
Truncating the sum at r terms defines a rank r matrix Sr = i=1 σi ui viT . Approximating A with Sr
Pk
Pk
leaves an error of Er = A − Sr = i=r+1 σi ui viT with |Er |2 = i=r+1 σi2 and |A|2 = |Sr |2 + |Er |2 . The
sum Sr is the result of making successive best rank one approximations. But there is a stronger result:
Sr is actually the best rank r approximation possible. Proofs of this assertion can be found in Lawson
and Hanson [14, pages 23–26] and Leon [16, pages 422–424].
The properties of reduced rank approximations can now be used to clarify a remark made earlier.
In the example showing the advantage of the SVD in least squares problems, the matrix A had three
independent columns, but the final column differed from a linear combination of the other two by a
small random vector. Subtracting that small random vector from the third column produces a rank
two matrix B that closely approximates A. In fact, |A − B| is equal to the norm of the random vector,
expected to be about 5 × 10−8 . Of course, the minimal error for a rank two approximation to A must
be precisely σ3 , so we can conclude that σ3 ≤ |A − B| ≈ 5 × 10−8 . This shows that the magnitude of the
random vector provides a bound on the least singular value of A. In particular, the order of magnitude
of the least singular value can be controlled by choosing the random vector appropriately. This is what
allowed us to construct A with a prescribed condition number in the least squares example.
In practice, the SVD can be used to choose the rank r, as well as the best rank r approximation to
A. Note that with r terms, the SVD outer product expansion results in a relative error
v
u Pk
2
|Er | u
i=r+1 σi
=t P
k
2
|A|
i=1 σi
Typically, the value of r is chosen to reduce this relative error to some prespecified threshhold. There
is a nice visual example in [17] of this idea, used to approximate an image of a surface in R3 . Image
20
processing is also discussed in [2] and [12, pages 222–224], the latter including an interesting illustration
involving fingerprints. A related discussion of the use of the SVD in cryptographic analysis appears in
[19]. For a completely different application of reduced rank approximations, see [6] which employs the
best rank 2 approximation of a matrix in connection with a data visualization technique.
We conclude this section with a detailed example of the SVD and reduced rank approximations.
This example was developed by K. Kirby [13].
The image shown in Fig. (6) is a representation of a 24 × 24 matrix A whose entries are all either 0
or 1. The image is created using a rectangular grid of the same dimensions, with each cell in the grid
Figure 6: A 24 × 24 image
colored black (if the corresponding matrix entry is 0) or white (if the entry is 1). The first 16 singular
values for this matrix are shown to four decimal places in the table below, the remaining singular values
21
were computed as zero to four decimal places.
9.5403
2.7385
1.4207
0.7479
6.6288
2.2023
1.2006
0.6744
5.6369
1.5835
0.9905
0.6122
3.4756
1.5566
0.9258
0.4698
Now suppose we adopt an accuracy threshhold of 90%. That would mean we wish
a reduced
qPto choose
r 2 P16 2
rank approximation with error no more than 10% of |A|. Define e(r) = 1 −
1 σi /
1 σi . This
gives the relative error for a sum of the first r terms of the SVD outer product expansion. Computing
e(2) = .18 and e(3) = .09, we see that three terms of the expansion are needed to achieve an error of 10%
or less. The result of using just three terms of the series is displayed as an image in Fig. (7). Here the
Figure 7: Rank 3 approximation
numerical entries of the matrix are displayed as gray levels. Similarly, setting a threshhold of 95% would
lead us to use a rank 5 approximation to the original matrix. That produces the image shown in Fig. (8),
in which one can recognize the main features of the original image. In fact, by simply rounding the entries
22
Figure 8: Rank 5 approximation
of the rank 5 approximation to the nearest integer, the original image is almostPperfectly restored, as
shown in Fig. (9). Observe in this regard that the error matrix E5 has |E5 |2 = i>5 σi2 = 16.7. Thus,
the average value of the squares of the entries of E5 is 16.7/242 = .029 and we might estimate that the
entries themselves ought to be around .17. Since this value is well below .5, it is not surprising that
Fig. (9) is nearly identical to the original image.
There is an intriguing analogy between reduced rank approximations and Fourier analysis. Particularly in the discrete case, Fourier analysis can be viewed as representing a data vector relative to a special
orthogonal basis. The basis elements are envisioned as pure vibrations, that is, sine and cosine functions,
at different frequencies. The Fourier decomposition thus represents the input data as a superposition
of pure vibrations with the coefficients specifying the amplitude of each constituent frequency. Often,
there are a few principal frequencies that account for most of the variability in the original data. The
remaining frequencies can be discarded with little effect. The reduced rank approximations based on
the SVD are very similar in intent. However, the SVD captures the best possible basis vectors for the
particular data observed, rather than using one standard basis for all cases. For this reason, SVD based
23
Figure 9: Rank 5 approximation, rounded to integers
reduced rank approximation can be thought of as an adaptive generalization of Fourier analysis. The
most significant vibrations are adapted to the particular data that appear.
A Computational Algorithm for the SVD
The applicability of the SVD is a consequence of its theoretical properties. In practical applications, the
software that calculates the SVD is treated as a black box: we are satisfied that the results are accurate,
and content to use them without worrying about how they were derived. However, peek into the box
and you will be rewarded with a glimpse of an elegant and powerful idea: implicit matrix algorithms.
The basic idea behind one of the standard algorithms for computing the SVD of A depends on the close
connection to the EVD of AT A. As the algorithm proceeds, it generates a sequence of approximations
Ai = Ui Σi ViT to the correct SVD of A. The validity of the SVD algorithm can be established by showing
that after each iteration, the product ATi Ai is just what would have been produced by the corresponding
iteration of a well known algorithm for the EVD of AT A. Thus, the convergence properties for the SVD
24
algorithm are inferred from those of the EVD algorithm, although AT A is never computed and the EVD
algorithm is never performed. From this perspective, the SVD algorithm can be viewed as an implicit
algorithm for the EVD of AT A. It provides all the information needed to contruct the EVD by operating
directly on A. Indeed, the operations on A are seen to implicitly perform the EVD algorithm for AT A,
without ever explicitly forming AT A.
Implicit algorithms are a topic of current research interest. References [1] and [4] describe some
implicit algorithms for computations other than the SVD, and suggest sources for further reading. A
detailed account of the SVD algorithm is found in [8, Sec. 8.3]. In the notes and references there, several
additional citations and notes about the history of the algorithm are also given. In [7] there is a more
compact (though less general) development that makes the connection to the QR algorithm more direct.
It should also be noted that the there are alternatives to the algorithm described above. One alternative
that has significance in some kinds of parallel processing is due to Jacobi, see [8, Sec. 8.4]. Another
interesting alternative uses a rank 1 modification to split an SVD problem into two problems of lower
dimension, the results of which can be used to find the SVD of the original problem. This method is
described in [11].
Conclusion
The primary goal of this paper is to bring the SVD to the attention of a broad audience. The theoretical
properties have been described, and close connections were revealed between the SVD and standard
topics in the first linear algebra course. Several applications of the SVD were mentioned, with a detailed
discussion of two: least squares problems and reduced rank estimation. The computational algorithm
for the SVD was also briefly mentioned.
In emphasizing the main themes of the subject, it has often been necessary to omit interesting details
and limit the presentation to general ideas. The reader is encouraged to consult the references for a
more thorough treatment of the many aspects of this singularly valuable decomposition.
Acknowledgments: Many thanks to Editor Bart Braden for significant contributions to this paper. Thanks also to
Professor Kevin Kirby for mathematica files used in the example of reduced rank approximation.
References
[1] Gary E. Adams, Adam W. Bojanzcyk, and Franklin T. Luk, Computing the PSVD of two 2 × 2
triangular matrices, SIAM J. Matrix Anal. Appl. vol 15, no 2, pp 366 – 382, April 1994.
[2] Harry C. Andrews and Claude L. Patterson, Outer Product Expansions and their uses in digital
image processing, The Amer Math Monthly, pp 1-13, January 1975.
[3] S. J. Blank, Nishan Krikorian, and David Spring, A geometrically inspired proof of the singular
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